The zeta function for the triangular potential

The zeta functions for the Schrodinger equation with a triangular potential are investigated. Values of the zeta functions are computed using both the Weierstrass factorization theorem and analytic continuation via contour integration. The results were found to be consistent where the domains of the two methods overlap. Analytic continuation is used to compute the values of the zeta functions at zero and the negative integers and explore the pole structure (and residue values) as well as the value of the slopes at the origin. Those results are used for the computation of the trace and determinant of the associated Hamiltonians.